Approximation algorithms for solving multi-objective optimization problems

This paper tries to cover the main aspects/properties related to scheduling problems, approximation algorithms, and multi-objective combinatorial optimization. Then, we try to describe the main techniques that can be used to solve such problems. In this paper, the reviews results relate to multi-objective optimization problems, exact and approximation search, with the aim of getting all Pareto optimal solutions for some NP-hard problems

Distance and consensus for preference relations corresponding to ordered partitions

Ranking is an important part of several areas of contemporary research, including social sciences, decision theory, data analysis and information retrieval. The goal of this paper is to align developments in quantitative social sciences and decision theory with the current thought in Computer Science, including a few novel results. Specifically, we consider binary preference relations, the so-called weak orders that are in one-to-one correspondence with rankings. We show that the conventional symmetric difference distance between weak orders, considered as sets of ordered pairs, coincides with the celebrated Kemeny distance between the corresponding rankings, despite the seemingly much simpler structure of the former. Based on this, we review several properties of the geometric space of weak orders involving the ternary relation “between”, and contingency tables for cross-partitions. Next, we reformulate the consensus ranking problem as a variant of finding an optimal linear ordering, given a correspondingly defined consensus matrix. The difference is in a subtracted term, the partition concentration, that depends only on the distribution of the objects in the individual parts. We apply our results to the conventional Likert scale to show that the Kemeny consensus rule is rather insensitive to the data under consideration and, therefore, should be supplemented with more sensitive consensus schemes

Encompassing the Work-Life Balance into Early Career Decision-Making of Future Employees Through the Analytic Hierarchy Process

The paper presents the results of ranking of the significance of quality of life determinants by University students that are starting professional activities. Research methodology: literature review; elaboration of an AHP decision-making model; two-stage expert selection; significance rankings by experts and a graphical and descriptive presentation of obtained results. Research sample: 14 experts out of almost 200 University students. Research outcome: a decision-making model that aims at maximizing the life satisfaction of future employees as a function of their individual assessments of significance of particular determinants of quality of life. Research implications: a more accurate adaptation to trends on the labor market and creation of new business models. Research limitation: narrowing the group of experts to University students. Value added of the research: better motivated employees with a satisfactory level of work-life balance will contribute to an increase of societal satisfaction level

Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: The application of evidence theory

AbstractWe consider multicriteria decision problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be uncertain and/or imprecise and is provided by one or several experts. We model this evaluation as a basic belief assignment (BBA). In order to compare the different pairs of alternatives according to each criterion, the concept of first belief dominance is proposed. Additionally, criteria weights are also expressed by means of a BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical example

A decision rule based on goal programming and one-stage models for uncertain multi-criteria mixed decision making and games against nature

This paper is concerned with games against nature and multi-criteria decision making under uncertainty along with scenario planning. We focus on decision problems where a deterministic evaluation of criteria is not possible. The procedure we propose is based on weighted goal programming and may be applied when seeking a mixed strategy. A mixed strategy allows the decision maker to select and perform a weighted combination of several accessible alternatives. The new method takes into consideration the decision maker’s preference structure (importance of particular goals) and nature (pessimistic, moderate or optimistic attitude towards a given problem). It is designed for one-shot decisions made under uncertainty with unknown probabilities (frequencies), i.e. for decision making under complete uncertainty or decision making under strategic uncertainty. The procedure refers to one-stage models, i.e. models considering combinations of scenarios and criteria (scenario-criterion pairs) as distinct meta-attributes, which means that the novel approach can be used in the case of totally independent payoff matrices for particular targets. The algorithm does not require any information about frequencies, which is especially desirable for new decision problems. It can be successfully applied by passive decision makers, as only criteria weights and the coefficient of optimism have to be declared

Une méthode de tri multicritère multi-périodes pour la sélection de projet en contexte d'incertitude

RÉSUMÉ: Dans les dernières années, le gouvernement du Québec a souligné l'importance de la prise de décision dans un contexte de développement durable et de lutte contre les changements climatiques. L'évaluation des projets dans ce contexte devrait prendre en considération l'équilibre entre les critères économiques, sociaux et environnementaux à court, moyen et long terme. De plus, ces évaluations peuvent être imprécises et tâchées d'incertitude. Les problèmes de décision dans ce contexte sont complexes et caractérisés par les trois aspects suivants, à savoir l'aspect multicritère, l'aspect temporel et l'incertitude. Or, la plupart des méthodes multicritères sont statiques et seules quelques rares méthodes traitent l'aspect temporel des évaluations. En effet, des recherches récentes ont développé des méthodes multicritères multi-périodes de rangement mais au meilleur de notre connaissance, aucune méthode de tri multicritère multi-périodes ne fut développée à date. L'objectif de ce mémoire est de proposer une méthode de tri multicritère multi-périodes dans un contexte d'incertitude pour l'évaluation de la durabilité des projets. La méthode proposée est constituée de deux phases d'agrégation multicritère et d'agrégation multi-périodes. La première phase consiste à conduire les simulations Monte Carlo et à appliquer la méthode SMAA-Tri pour affecter à chaque période le projet à une des catégories prédéfinies. Ensuite, la phase d'agrégation multi-périodes propose d'agréger les résultats obtenus dans chaque période pour arriver à une affectation à la fois multicritère et multi-périodes. La méthode proposée a été appliquée dans le contexte d'aménagement forestier durable. Un projet d'aménagement spécifique qui consiste à implanter un plan de protection spécifique pour l'habitat du caribou a été trié selon un ensemble de critères évalués sur l'horizon de régénération de la forêt de 150 ans. L'incertitude a été simulée par 10000 simulations Monte Carlo à chacune des 30 périodes. Les résultats de cette application démontrent que la méthode proposée permet de généraliser la méthode SMAA Tri au contexte multi-périodes et aboutit à des résultats intéressants. -- Mot(s) clé(s) en français : Sélection de projet, Méthodes de tri multicritère, évaluations multi-périodes, Monte Carlo, incertitude, développement durable. -- ABSTRACT: In the last years, the government of Quebec emphasized sustainable and robust decision making in the context of climate change. Projects evaluation in this context must take into consideration the balance between economic, social and environmental criteria, over the short, medium and long term. Furthermore, decision criteria may be imprecise or uncertain. Decision-making problems in this context are complex and characterized by multi-criteria, temporal and uncertainty aspects. Yet, the majority of the multi-criteria methods are static and only few methods deal with temporal evaluations. In fact, recent studies proposed multi-criteria multi-period ranking methods but to the best of our knowledge, there is no multi-criteria multi-period sorting method proposed yet. The general objective of this research is to propose a multi-criteria multi-period sorting method in the context of uncertainty to be used for sustainability evaluations of projects. The proposed method is composed of two phases, the multi-criteria aggregation phase, and the multi-period aggregation phase. The aggregation phase consists of conducting the Monte-Carlo Simulations and applying the SMAA-TRI method at each period in order to sort the project in one of the predefined categories. Then, the multi-period aggregation proposes to aggregate the results obtained at each period in order to get a global sorting result. The proposed method is applied in the context of sustainable forest management. A particular project of forest management, that aims to implement a specific protection plan for the caribou habitat, is sorted according to a set of criteria evaluated over the regeneration forest horizon of 150 years. Uncertainty has been simulated with 10 000 Monte-Carlo simulations over 30 periods. The results of this application show that the proposed method generalizes the SMAA-TRI method to the multi-period context and provides interesting results. -- Mot(s) clé(s) en anglais : Project selection, multi-criteria sorting methods, multi-period evaluations, Monte Carlo, uncertainty, sustainable development

A Decision Rule for Uncertain Multi-Criteria Pure Decision Making and Independent Criteria

The paper is concerned with multi-criteria decision-making under uncertainty with scenario planning. This topic has been explored by many researchers since almost all real-world decision problems contain multiple conflicting criteria and a deterministic evaluation of criteria is often impossible. We propose a procedure for uncertain multi-objective optimization which can be applied when seeking a pure strategy. A pure strategy, as opposed to a mixed strategy, allows the decision-maker to select and perform only one accessible alternative. The new approach takes into account the decision-maker’s preference structure (importance of particular goals) and nature (pessimistic, moderate or optimistic attitude towards a given problem). It is designed for one-shot decisions made under uncertainty with unknown probabilities (frequencies), see decision-making under complete uncertainty or decision-making under strategic uncertainty. The novel approach can be used in the case of totally independent payoff matrices for particular targets.This research is financed by the National Science Center in Poland (project registration number: 2014/15/D/HS4/00771)[email protected] of Informatics and Electronic Economy, Poznań University of Economics and BusinessAghdaie M. H., Zolfani S. H., Zavadskas E. K., 2013, Market Segment Evaluation and Selection Based on Application of Fuzzy AHP and COPRAS-G Methods, “Journal of Business Economics and Management” vol. 14(1).Ben Amor S., Jabeur K., Martel J., 2007, Multiple Criteria Aggregation Procedure for Mixed Evaluations, “European Journal of Operational Research”, vol. 181, iss. 3.Benzion U., Cohen Y., Shavit T., 2010, The Newsvendor Problem with Unknown Distribution, “Journal of the Operational Research Society” iss. 6(6), DOI 10.1057/ jors.2009.56Besbes O., Muharremoglu A., 2013, On Implications of Censoring Demand in the Newsvendor Problem, “Management Science”, vol. 59, iss. 6, DOI 10.2139/ssrn.1983270Bieniek M., 2016, Bicriteria Optimization in the Newsvendor Problem with Exponentially Distributed Demand, “Multiple Criteria Decision Making”, vol. 11.Caplan B., 2001, Probability, Common Sense, and Realism: A Reply to Hulsmann and Block, “The Quarterly Journal of Austrian Economics”, vol. 4, no. 2.Carnap R., 1950, Logical Foundations of Probability, University Press, Chicago.De Finetti B., 1975, Theory of Probability. A Critical Introductory Treatment, Wiley, London.Dominiak C., 2006, Multicriteria Decision aid under Uncertainty, “Multiple Criteria Decision-Making’ 05”.Dominiak C., 2009, Multi-Criteria Decision Aiding Procedure under Risk and Uncertainty, “Multiple Criteria Decision Making’ 08”.Dubois D., Prade H., 2001, Possibility Theory, Probability Theory and Multiple-Valued Logics: A Clarification, “Annals of Mathematics and Artificial Intelligence” vol. 32.Dubois D., Prade H., 2012, Gradualness, Uncertainty and Bipolarity: Making Sense of Fuzzy Sets, “Fuzzy Sets and Systems”, vol. 192.Durbach I. N., 2014, Outranking under Uncertainty Using Scenarios, “European Journal of Operational Research”, vol. 232, iss. 1.Durbach I. N., Stewart T. J., 2012, Modeling Uncertainty in Multi-Criteria Decision Analysis, “European Journal of Operational Research”, vol. 223, iss. 1.Eiselt H. A., Marianov V., 2014, Multi-Criteria Decision-Making under Uncertainty: A Visual Approach, “International Transactions in Operational Research”, vol. 21, iss. 4.Fishburn P.C., 1984, Foundations of Risk Measurement. I. Risk or Probable Loss, “Management Science”, vol. 30.Frechet M., 1938, The Diverse Definitions of Probability. »Lecture at the fourth International Congress for the Unity of Science« Erkenntnis.Gaspars H., 2007, Ressource Allocation under Uncertainty: Choice Models and Computational Procedures [Alokacja zasobu w warunkach niepewności: modele decyzyjne i procedury obliczeniowe]. „Operations Research and Decisions”, vol. 1 (in Polish).Gaspars-Wieloch H., 2014a, A Hybrid of the Hurwicz and Bayes Rules in Decision- Making under Uncertainty [Propozycja hybrydy reguł Hurwicza i Bayesa w podejmowaniu decyzji w warunkach niepewności]. T. Trzaskalik (Ed.) „Modelowanie Preferencji a Ryzyko 2014. Studia Ekonomiczne. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Katowicach”, 178, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice (in Polish).Gaspars-Wieloch H., 2014b, On a Decision Rule for Mixed Strategy Searching under Uncertainty on the Basis of the Coefficient of Optimism, “Procedia – Social and Behavioral Sciences”, vol. 110.Gaspars-Wieloch H., 2014c, Modifications of the Hurwicz’s Decision Rules, “Central European Journal of Operations Research”, vol. 22, iss. 4.Gaspars-Wieloch H., 2014d, The Use of a Modification of the Hurwicz’s Decision Rule in Multicriteria Decision-Making under Complete Uncertainty, “Business, Management and Education”, vol. 12, iss. 2.Gaspars-Wieloch H., 2015a, Modifications of the Omega Ratio for Decision- Making under Uncertainty, “Croatian Operational Research Review”, vol. 6, no. 1.Gaspars-Wieloch H., 2015b, On a Decision Rule Supported by a Forecasting Stage Based on the Decision-Maker’s Coefficient of Optimism, “Central European Journal of Operations Research”, vol. 23, iss. 3, DOI 10.1007/s10100-014-0364-5Gaspars-Wieloch H., 2015c, A Decision Rule for Uncertain Multicriteria Mixed Decision-Making Based on the Coefficient of Optimism, “Multiple Criteria Decision Making ’15”, University of Economics in Katowice.Gaspars-Wieloch H., 2015d, On a Decision Rule for Searching an Optimal Pure Strategy in Uncertain Multi-Criteria Decision-Making [O regule decyzyjnej wspierającej wielokryterialne poszukiwanie optymalnej strategii czystej w warunkach niepewności], “Studia Ekonomiczne. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Katowicach”, University of Economics in Katowice (in Polish).Gaspars-Wieloch H., 2016a, Newsvendor Problem under Complete Uncertainty – A Case of Innovative Products, “Central European Journal of Operations Research”, DOI 10.1007/s10100- 016-0458-3.Gaspars-Wieloch H., Michalska E., 2016b, On two Applications of the Omega Ratio: MaxΩmin and Omega(H+B), “Research Papers of Wrocław University of Economics” (Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu), no. 446, Metody i zastosowania badań operacyjnych. Wrocław, 21-36 DOI 10.15611/pn.2016.446.02Gaspars-Wieloch H., 2017, A Decision Rule Based on Goal Programming and One-Stage Models for Uncertain Multi-Criteria Mixed Decision-Making and Games against Nature, “Croatian Operational Research Review”, vol. 8, no. 1.Ginevičius R., Zubrecovas V., 2009, Selection of the Optimal Real Estate Investment Project Basing on Multiple Criteria Evaluation Using Stochastic Dimensions, “Journal of Business Economics and Management”, vol. 10, iss. 3.Goodwin P., Wright G., 2001, Enhancing Strategy Evaluation in Scenario Planning: A Role for Decision Analysis, “Journal of Management Studies”, vol. 38, iss. 1.Guney S., Newell B.R., 2015, Overcomming Ambiguity Aversion through Experience, “Journal of Behavioral Decision Making”, vol. 28, iss. 2.Guo P., 2011, One-Shot Decision Theory, “IEEE Transactions on Systems, Man, and Cybernetics”, Part A, vol. 41, iss. 5.Guo P., Ma X. 2014 Newsvendor Models for Innovative Products with One-Shot Decision Theory, “European Journal of Operational Research”, vol. 239, DOI 10.1016/j.ejor.2014.05.028Hau R., Pleskac T.J., Hertwig R., 2009, Decisions from Experience and Statistical Probabilities: Why they Trigger Different Choices than a priori Probabilities?, “Journal of Behavioral Decision Making”, DOI 10.1002/bdm.665Hopfe C. J., Augenbroe G.L.M., Hensen J.L.M., 2013, Multicriteria Decision-Making under Uncertainty in Building Performance Assessment, “Building and Environment”, vol. 69.Hurwicz L., 1952, A Criterion for Decision-Making under Uncertainty, “Technical Report”, 355, Cowles Commission.Janjic A., Andjelkovic A., Docic M., 2013, Multiple Criteria Decision-Making under Uncertainty Based on Stochastic Dominance, “Proceedings of the 2013 International Conference on Applied Mathematics and Computational Methods in Engineering” 16-19 July 2013, Rhodes Island, Greece.Kahneman D., 2011, Thinking, Fast and Slow, Farrar, Straus & Giroux.Kaplan S., Barish N.N., 1967, Decision-Making Allowing for Uncertainty of Future Investment Opportunities, “Management Science”, vol. 13, iss. 10.Knight F.H., 1921, Risk, Uncertainty, Profit, Hart. Boston MA, Schaffner & Marx, Houghton Mifflin Co.Kolmogorov A. N., 1933, Grundbegriffe der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin.Korhonen A., 2001, Strategic Financial Management in a Multinational Financial Conglomerate: A Multiple Goal Stochastic Programming Approach, “European Journal of Operational Research”, vol. 128.Lee Y.-H., 2012, A Fuzzy Analytic Network Process Approach to Determining Prospective Competitive Strategy in China: A Case Study for Multinational Biotech Pharmaceutical Enterprises, “Journal of Business Economics and Management”, vol. 13, iss. 1.Liu Y., Fan Z., Hang Y., 2011, A Method for Stochastic Multiple Criteria Decision- Making Based on Dominance Degrees, “Information Sciences”, vol. 181, iss. 19.Liu B., 2007, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin.Liu B., 2009, Some Research Problems in Uncertainty Theory, “Journal of Uncertain Systems”, vol. 3, no. 1.Lo M.C., Michnik J., 2010, An Evaluation Method Based on Multi-Attributes Analysis with Stochastic Dominances for Improving the Information Quality, “International Journal of Information Systems for Logistics and Management”, vol. 6.Michnik J., 2013, Scenario Planning+MCDA Procedure for Innovation Selection Problem, “Foundations of Computing and Decision Sciences”, 38(3).Mikhaidov L., Tsvetinov P. 2004 Evaluation of Services Using a Fuzzy Analytic Hierarchy Process, “Applied Soft Computing”, vol. 5, iss. 1.Millett S.M., 2009, Should Probabilities Be Used with Scenarios?, “Journal of Futures Studies”, 13(4).Montibeller G., Gummer H., Tumidei D. 2006 Combining Scenario Planning and Multi-Criteria Decision Analysis in Practice, “Journal of multi-criteria decision analysis” vol. 14.Ogryczak W., Śliwiński T., 2009, On Efficient WOWA Optimization for Decision Support under Risk, “International Journal of Approximate Reasoning”, vol. 50.Operations Research [Badania Operacyjne], 2008, W. Sikora (ed), Polskie Wydawnictwo Ekonomiczne, Warsaw (in Polish).Perez D.E, Hernandez J.G., Garcia M.J., Hernandez G.J., 2015, Hurwicz Method Modified and the Amplitude Model (TAM), In Delener et al. (Ed), GBATA 2015 Reading book, GBATA, USA.Piasecki K., 2016, Intuicyjne zbiory rozmyte jako narzędzie finansów behawioralnych [Intuitive Fuzzy Sets as a Tool of Behavioral Finances], edu-Libri (in Polish).Piegat A., 2010, Uncertainty of Probability, Workshop on Intuitionistic Fuzzy Sets and Generalized Nets.Popper K., 1959, The Propensity Interpretation of Probability, “The British Journal of the Philosophy of Science”, 10(37).Ram C., Montibeller G., Morton A., 2010, Extending the Use of Scenario Planning and MCDA for the Evaluation of Strategic Options, “Journal of Operational Research Society”, vol. 62, iss. 5.Ramík J., Hanclova J., Trzaskalik T., Sitarz S., 2008, Fuzzy Multiobjective Methods in Multistage Decision Problems, “Multiple Criteria Decision-Making ‘07”.Ramsey F., 1931, Truth and Probability, In The Foundations of Mathematics and other Logical Essays, by Frank Ramsey, Routledge, Kegan Paul, London.Ravindran A. R., 2008, Operations Research and Management Science Handbook, Boca Raton, London, New York, CRS Press.Shafer G., 1976, A Mathematical Theory of Evidence, Princeton University Press, Princeton.Sentz K., Ferson S., 2002, Combination of Evidence in Dempster-Shafer Theory, Sandia Report SAND2002-0835, April 2002, Albuquerque, NM.Silva R.C., 2016, A Multiobjective Approach to Solve Container Ship Loading Planning Problem with Uncertainty, Conference Handbook, 28th European Conference on Operational Research, EURO, The Association of European Operational Research Societies.Stewart T. J., 2005, Dealing with Uncertainties in MCDA, Multiple Criteria Decision Analysis: State-of-the-Art Surveys, “International Series in Operations Research & Management Science”, 78.Stirling W.C., 2003, Satisficing Games and Decision-Making. With Applications to Engineering and Computer Science, Cambridge University Press.Suo M.Q., Li Y.P., Huang G.H., 2012, Multicriteria Decision-Making under Uncertainty: An Advanced Ordered Weighted Averaging Operator for Planning Electric Power Systems, “Engineering Applications of Artificial Intelligence”, vol. 25, iss. 1.Troutt M.D., Pettypool M.D., 1989, On the Role of Mixed Strategies in the Elementary Decision Analysis and Related Decision-Support-System Treatments, “Journal of Operational Research Society”, vol. 40, iss. 6.Trzaskalik T., 2008, Introduction to Operations Research with Computer [Wprowadzenie do badań operacyjnych z komputerem], 2nd ed. Polskie Wydawnictwo Ekonomiczne, Warsaw (in Polish).Trzaskalik T., 2014, Multicriteria Decision Aiding [Wielokryterialne wspomaganie decyzji], Polskie Wydawnictwo Ekonomiczne, Warsaw (in Polish).Tsaur S., Chang T., Yen C., 2002, The Evaluation of Airline Service Quality by Fuzzy MCDM, “Tourism Management”, vol. 23, iss. 2.Urli B., Nadeau R. 2004 PROMISE/scenarios: An Interactive Method for Multiobjective Stochastic Linear Programming under Partial Uncertainty, “European Journal of Operational Research”, vol. 155, iss. 2.Van Lambalgen M. 1996 Randomness and Foundations of Probability: von Mises' Axiomatization of Random Sequences. Probability, Statistics and Game Theory, Papers in honor of David Blackwell, Institute for Mathematical Statistics.Von Mises L., 1949, Human Action: A Treatise on Economics, Yale University Press.Von Mises R, 1957, Probability, Statistics and Truth, The Macmillan Company, New York.von Neumann J., Morgenstern O., 1944, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New York.Wachowicz T., 2015, Methods of Multi-Criteria Decision Analysis in Quantitative Scientific Research [Metody wielokryterialnej analizy decyzyjnej w ilościowych badaniach naukowych], [in:] Podstawy metodologii badań w naukach o zarządzaniu, W. Czakon (ed.), Oficyna Wolters Kluwer business (in Polish)Walley P., 1991, Statistical Reasoning with Imprecise Probabilities, Chapman and Hall.Waters D., 2011, Supply Chain Risk Management. Vulnerability and Resilience in Logistics. Kogan Page.Weber M., 1987, Decision-Making with Incomplete Information, “European Journal of Operational Research” vol. 28.Wojewnik P., Szapiro T., 2010, Bireference Procedure FBI for Interactive Multicriteria Optimization with Fuzzy Coefficients, “Central European Journal of Economic Modelling and Econometrics”, vol. 2.Xu R., 2000, Fuzzy Least-Squares Priority Method in the Analytic Hierarchy Process, “Fuzzy Sets and Systems”, vol. 112, iss. 3.Yu C., 2002, A GP-AHP Method for Solving Group Decision-Making Fuzzy AHP Problems, “Computers and Operations Research”, vol. 29, iss. 14.Zadeh L., 1978, Fuzzy Sets as the Basis for a Theory of Possibility, “Fuzzy Sets and Systems” 1.Zio E., Pedroni N., 2013, Methods for Representing Uncertainty. A Literature Review. Apports de la recherche 2013-3, Risk Analysis. Les cahiers de la securite industrielle, FONCSI.3(87)779